Delay Equations Research Articles

Boolean Delay Equations: a dynamical approach to modeling complex systems

Boolean delay equations (BDEs) are equations with discrete variables evolving in continuous time. They serve as exploratory tools in the study of nonlinear and complex systems. In this review paper, we outline their formulation and illustrate their properties by application to a solid-earth problem and a climate one. The first problem is the seismotectonic description and prediction of earthquakes and of their clustering. The second one is that of the coupled atmosphere-ocean phenomenon of the El Niño-Southern Oscillation (ENSO). Both involve irregular behavior that is hard to predict, although some form of cyclicity is present in both, especially in ENSO. The paper concludes with broad perspectives on the further use of BDEs in the geosciences and elsewhere.

Delay Equations and Admissible Spaces

Delay Equations and Admissible Spaces

Differential Difference Equations in Trajectory Planning and Control for Differential Drive Robots: A Comprehensive Exploration

This paper provides a thorough investigation into the utilisation of differential difference equations (DDEs) for the purposes of trajectory planning and control in the context of differential drive four-wheeled robots. The inclusion of sensor and control delays is crucial when developing navigation strategies that are both resilient and efficient for robots functioning in dynamic environments. Differential delay equations (DDEs) provide a robust mathematical framework for representing the dynamics of such systems, facilitating precise and reliable path tracking. In order to demonstrate the versatility and practicality of Delay Differential Equations (DDEs), we investigate three distinct scenarios: straight-line path tracking, circular path tracking, and path planning with obstacle avoidance. These scenarios serve as demonstrations of how Delay Differential Equations (DDEs) facilitate the robot's ability to adjust its motion by utilising previous information, thereby ensuring a trajectory tracking process that is both smooth and stable. The trajectory plots provide a visual representation of the robot's path and effectively illustrate the successful completion of various navigation objectives. This study highlights the importance of dynamic differential equations (DDEs) in the advancement of intelligent and adaptable robotic systems, thereby paving the way for future progress in the realm of robotics and autonomous systems.

Exponential stabilization of stochastic quantum systems based on time-delay noise-assisted feedback

The exponential stabilization of stochastic quantum systems (SQS) under time-delay noise-assisted feedback is addressed based on the exponential stability of the stochastic differential delay equation (SDDE) in this paper. We show that if the stochastic differential equation (SDE), which covers the dynamical models of SQS under noise-assisted feedback, is exponentially stable, then the corresponding SDDE is also exponentially stable when the time-varying delay is bounded by a positive number, which is given in an implicit expression and can be computed numerically. On this basis, the results for the general SDDE are applied to the SQS under time-delay noise-assisted feedback, and the corresponding time-delay boundary is obtained. Furthermore, we show the time-delay boundary in numerical simulations.

Estimating the region of attraction of integral delay equations

Abstract We present estimates of the region of attraction of a class of non-linear integral delay equations. The approach follows the same lines as the Lyapunov method by the first approximation for estimating the attraction region of ordinary and delay differential equations, which uses Lyapunov functions or Lyapunov functionals associated with the linear part of the non-linear equations. The results are illustrated by application problems to the propagation of electric pulses in the heart dynamics and non-linear predictor control.

Admissibility via Induced Delay Equations

Admissibility via Induced Delay Equations

Partially-Observed Maximum Principle for Backward Stochastic Differential Delay Equations

Partially-Observed Maximum Principle for Backward Stochastic Differential Delay Equations

Explicit numerical approximations for SDDEs in finite and infinite horizons using the adaptive EM method: Strong convergence and almost sure exponential stability

In this paper we investigate explicit numerical approximations for stochastic differential delay equations (SDDEs) under a local Lipschitz condition by employing the adaptive Euler-Maruyama (EM) method. Working in both finite and infinite horizons, we achieve strong convergence results of the adaptive EM solution. We also obtain the order of convergence in finite horizon. In addition, we show almost sure exponential stability of the adaptive approximate solution for both SDEs and SDDEs.

Development and calibration of a multi-delay coherence imaging diagnostic on the MAST-U tokamak.

The MAST-U Super-X divertor provides the opportunity to study fusion plasma exhaust under novel conditions. However, in order to study these conditions, advanced diagnostics are required. Following the development of the MAST-U Multi-Wavelength Imaging (MWI) diagnostic, we present the installation of a multi-delay coherence imaging spectroscopy (CIS) system within the MAST-U MWI, along with modifications made to the MWI for effective operation. This diagnostic will measure either carbon ion flow velocities and temperatures or electron densities through Dγ emission. We have extended previously developed techniques for wavelength calibration to account for errors due to the misalignment of interferometer components. In addition, we have developed a comprehensive calibration procedure to account for the temperature dependence of the instrument's delays by fitting to a linearly modified version of the delay equation presented by Veiras etal. [Appl. Opt. 49(15), 2769 (2010)]. Together, these procedures reduce the cost and hardware complexity of implementing CIS instruments when compared to those that use in situ or tunable laser calibration systems, as calibrations can be generated to good accuracy using previously measured data.

On hierarchical competition through reduction of individual growth

We consider a population organised hierarchically with respect to size in such a way that the growth rate of each individual depends only on the presence of larger individuals. As a concrete example one might think of a forest, in which the incidence of light on a tree (and hence how fast it grows) is affected by shading by taller trees. The classic formulation of a model for such a size-structured population employs a first order quasi-linear partial differential equation equipped with a non-local boundary condition. However, the model can also be formulated as a delay equation, more specifically a scalar renewal equation, for the population birth rate. After discussing the well-posedness of the delay formulation, we analyse how many stationary birth rates the equation can have in terms of the functional parameters of the model. In particular we show that, under reasonable and rather general assumptions, only one stationary birth rate can exist besides the trivial one (associated to the state in which there are no individuals and the population birth rate is zero). We give conditions for this non-trivial stationary birth rate to exist and analyse its stability using the principle of linearised stability for delay equations. Finally, we relate the results to the alternative, partial differential equation formulation of the model.

Almost sure exponential stability and stochastic stabilization of impulsive stochastic differential delay equations

In this paper, we mainly study the almost sure exponential stability of impulsive stochastic differential delay equations (ISDDEs) with bounded variable delays. The main technique is to compare ISDDEs with corresponding impulsive stochastic differential equations (ISDEs) without delay, to obtain the upper bound τ∗ of delays that ISDDEs can maintain stability by accurate calculation. The results show that if the corresponding ISDEs are almost surely exponentially stable, then the ISDDEs are also almost surely exponentially stable as long as these delays are less than τ∗. In addition, the stability theory established in this paper can be applied to noise stabilization based on sampled-data observations of a class of unstable impulsive systems. Taking impulsive Lurie systems (ILSs) as an example, we discuss the design of noise stabilization strategies.

Two Different Analytical Approaches for Solving the Pantograph Delay Equation with Variable Coefficient of Exponential Order

The pantograph equation is a basic model in the field of delay differential equations. This paper deals with an extended version of the pantograph delay equation by incorporating a variable coefficient of exponential order. At specific values of the involved parameters, the exact solution is obtained by applying the regular Maclaurin series expansion (MSE). A second approach is also applied on the current model based on a hybrid method combining the Laplace transform (LT) and the Adomian decomposition method (ADM) denoted as (LTADM). Although the MSE derives the exact solution in a straightforward manner, the LTADM determines the solution in a closed series form which is theoretically proved for convergence. Further, the accuracy of such a closed-form solution is examined through various comparisons with the exact solution. For validation, the residual errors are calculated and displayed in graphs. The results show that the solution obtained utilizing the LTADM is in full agreement with the exact solution using only a few terms of the closed-form series solution. Moreover, it is found that the residual errors tend to zero, which reflects the effectiveness of the LTADM. The present approach may merit further extension by including other types of linear delay differential equations with variable coefficients.

Existence of strong solutions for one-dimensional reflected mixed stochastic delay differential equations

Abstract Relying on the pathwise uniqueness property, we prove existence of the strong solution of a one-dimensional reflected stochastic delay equation driven by a mixture of independent Brownian and fractional Brownian motions. The difficulty is that on the one hand we cannot use the fixed-point and contraction mapping methods because of the stochastic and pathwise integrals, and on the other hand the non-continuity of the Skorokhod map with respect to the norms considered.

Non-polynomial spline approach for solving system of singularly perturbed delay differential equations of large delay

ABSTRACT A computational technique for analysing coupled system of singularly perturbed convection-diffusion delay differential equations with large delay is presented in this study. We solve this problem using non-polynomial spline technique and arrive at a system that can potentially be solved. To show the theoretical investigation, an example is solved for different perturbation parameter values, and the computational results are shown in tables. The model we have proposed is of first order convergent, and this technique indicated that the acquired findings were more accurate than earlier results.

Application of an Ansatz Method on a Delay Model With a Proportional Delay Parameter

Application of an Ansatz Method on a Delay Model With a Proportional Delay Parameter

Comments to ‘Stability of traveling waves for deterministic and stochastic delayed reaction-diffusion equation based on phase shift’

In Liu et al. (2023), the authors set out to prove orbital stability for traveling waves in a (stochastic) delay reaction–diffusion equation. For the deterministic proof, they follow a phase-tracking technique as developed by Zumbrun and Howard (1998) [1] and for the stochastic version, they use a stochastic phase-tracking technique as developed by Hamster and Hupkes (2019, 2020a,b) [2,3] (which in turn is based on the aforementioned technique by Howard and Zumbrun). Note that the authors do not acknowledge the origins of their techniques.In the deterministic case, stability results are known (Mei, 2009) [4], but the application of phase-tracking to this problem appears to be new. However, there is a very subtle interplay between the time-dependent phase and the delay in the delay equation which is not accounted for, nor is explained or proved if the chosen phase has the same properties in the general framework of delay semigroups as for regular semigroups.The proof of the stochastic case of course has the same issues, but let us assume that the issues can be resolved, then there are still several issues with the proof. Especially, the phase tracking introduces terms that depend on the gradient of the solution which must be bounded in the H1 norm. This introduces singularities in the bounds that do not show in these proofs.

Well-posedness and order preservation for neutral type stochastic differential equations of infinite delay with jumps

<abstract><p>In this work, we are concerned with the order preservation problem for multidimensional neutral type stochastic differential equations of infinite delay with jumps under non-Lipschitz conditions. By using a truncated Euler-Maruyama scheme and adopting an approximation argument, we have developed the well-posedness of solutions for a class of stochastic functional differential equations which allow the length of memory to be infinite, and the coefficients to be non-Lipschitz and even unbounded. Moreover, we have extended some existing conclusions on order preservation for stochastic systems to a more general case. A pair of examples have been constructed to demonstrate that the order preservation need not hold whenever the diffusion term contains a delay term, although the jump-diffusion coefficient could contain a delay term.</p></abstract>

Advances in mathematical analysis for solving inhomogeneous scalar differential equation

<p>This paper considered a functional model which splits to two types of equations, mainly, advance equation and delay equation. The advance equation was solved using an analytical approach. Different types of solutions were obtained for the advance equation under specific conditions of the model's parameters. These solutions included the polynomial solutions of first and second degrees, the periodic solution and the hyperbolic solution. The periodic solution was invested to establish the analytical solution of the delay equation. The characteristics of the solution of the present model were discussed in detail. The results showed that the solution was continuous in the domain of the problem, under a restriction on the given initial condition, while the first derivative was discontinuous at a certain point and lied within the domain of the delay equation. In addition, some existing results in the literature were recovered as special cases of the current ones. The present successful analysis can be further generalized to include complex functional equations with an arbitrary function as an inhomogeneous term.</p>

Analysis of solution of fractional summation-difference equation of finite delay in cone metric space

Analysis of solution of fractional summation-difference equation of finite delay in cone metric space

Periodic solutions in reversible systems in second order systems with distributed delays

<abstract><p>In this paper, we study the existence and multiplicity of periodic solutions to a class of second-order nonlinear differential equations with distributed delay. Under assumptions that the nonlinearity is odd, differentiable at zero and satisfies the Nagumo condition, by applying the equivariant degree method, we prove that the delay equation admits multiple periodic solutions. The results are then illustrated by an example.</p></abstract>